Chapter 5: Inexact Reasoning

Chapter 5:Inexact Reasoning Expert Systems: Principles and Programming, Fourth Edition

Objectives • Explore the sources of uncertainty in rules • Analyze some methods for dealing with uncertainty • Learn about the Dempster-Shafer theory • Learn about the theory of uncertainty based on fuzzy logic • Discuss some commercial applications of fuzzy logic Expert Systems: Principles and Programming, Fourth Edition

Uncertainty and Rules • We have already seen that expert systems can operate within the realm of uncertainty. • There are several sources of uncertainty in rules: • Uncertainty related to individual rules • Uncertainty due to conflict resolution • Uncertainty due to incompatibility of rules Expert Systems: Principles and Programming, Fourth Edition

Figure 5.1 Major Uncertainties in Rule-Based Expert Systems Expert Systems: Principles and Programming, Fourth Edition

Figure 5.2 Uncertainties in Individual Rules Expert Systems: Principles and Programming, Fourth Edition

Figure 5.3 Uncertainty Associated with the Compatibilities of Rules Expert Systems: Principles and Programming, Fourth Edition

Figure 5.4 Uncertainty Associated with Conflict Resolution Expert Systems: Principles and Programming, Fourth Edition

Goal of Knowledge Engineer • The knowledge engineer endeavors to minimize, or eliminate, uncertainty if possible. • Minimizing uncertainty is part of the verification of rules. • Verification is concerned with the correctness of the system’s building blocks – rules. Expert Systems: Principles and Programming, Fourth Edition

Verification vs. Validation • Even if all the rules are correct, it does not necessarily mean that the system will give the correct answer. • Verification refers to minimizing the local uncertainties. • Validation refers to minimizing the global uncertainties of the entire expert system. • Uncertainties are associated with creation of rules and also with assignment of values. Expert Systems: Principles and Programming, Fourth Edition

Ad Hoc Methods • The ad hoc introduction of formulas such as fuzzy logic to a probabilistic system introduces a problem. • The expert system lacks the sound theoretical foundation based on classical probability. • The danger of ad hoc methods is the lack of complete theory to guide the application or warn of inappropriate situations. Expert Systems: Principles and Programming, Fourth Edition

Sources of Uncertainty • Potential contradiction of rules – the rules may fire with contradictory consequents, possibly as a result of antecedents not being specified properly. • Subsumption of rules – one rules is subsumed by another if a portion of its antecedent is a subset of another rule. Expert Systems: Principles and Programming, Fourth Edition

Uncertainty in Conflict Resolution • There is uncertainty in conflict resolution with regard to priority of firing and may depend on a number of factors, including: • Explicit priority rules • Implicit priority of rules • Specificity of patterns • Recency of facts matching patterns • Ordering of patterns • Lexicographic • Means-Ends Analysis • Ordering that rules are entered Expert Systems: Principles and Programming, Fourth Edition

Uncertainty • When a fact is entered in the working memory, it receives a unique timetag – indicating when it was entered. • The order that rules are entered may be a factor in conflict resolution – if the inference engine cannot prioritize rules, arbitrary choices must be made. • Redundant rules are accidentally entered / occur when a rule is modified by pattern deletion. Expert Systems: Principles and Programming, Fourth Edition

Uncertainty • Deciding which redundant rule to delete is not a trivial matter. • Uncertainty arising from missing rules occurs if the human expert forgets or is unaware of a rule. • Data fusion is another cause of uncertainty – fusing of data from different types of information. Expert Systems: Principles and Programming, Fourth Edition

Certainty Factors • Another method of dealing with uncertainty uses certainty factors, originally developed for the MYCIN expert system. Expert Systems: Principles and Programming, Fourth Edition

Difficulties with Bayesian Method • The Bayesian method is useful in medicine / geology because we are determining the probability of a specific event (disease / location of mineral deposit), given certain symptoms / analyses. • The problem is with the difficulty / impossibility of determining the probabilities of these givens – symptoms / analyses. • Evidence tends to accumulate over time. Expert Systems: Principles and Programming, Fourth Edition

Belief and Disbelief • Consider the statement: “The probability that I have a disease plus the probability that I do not have the disease equals one.” • Now, consider an alternate form of the statement: “The probability that I have a disease is one minus the probability that I don’t have it.” Expert Systems: Principles and Programming, Fourth Edition

Belief and Disbelief • It was found that physicians were reluctant to state their knowledge in the form: “The probability that I have a disease is one minus the probability that I don’t have it.” • Symbolically, P(H|E) = 1 – P(H’|E), where E represents evidence Expert Systems: Principles and Programming, Fourth Edition

Likelihood of Belief / Disbelief • The reluctance by the physicians stems from the likelihood of belief / disbelief – not in the probabilities. • The equation, P(H|E) = 1 – P(H’|E), implies a cause-and-effect relationship between E and H. • The equation implies a cause-and-effect relationship between E and H’ if there is a cause-and-effect between E and H. Expert Systems: Principles and Programming, Fourth Edition

Measures of Belief and Disbelief • The certainty factor, CF, is a way of combining belief and disbelief into a single number. • This has two uses: • The certainty factor can be used to rank hypotheses in order of importance. • The certainty factor indicates the net belief in a hypothesis based on some evidence. Expert Systems: Principles and Programming, Fourth Edition

Certainty Factor Values • Positive CF – evidence supports the hypothesis • CF = 1 – evidence definitely proves the hypothesis • CF = 0 – there is no evidence or the belief and disbelief completely cancel each other. • Negative CF – evidence favors negation of the hypothesis – more reason to disbelieve the hypothesis than believe it Expert Systems: Principles and Programming, Fourth Edition

Threshold Values • In MYCIN, a rule’s antecedent CF must be greater than 0.2 for the antecedent to be considered true and activate the rule. • This threshold value minimizes the activation of rules that only weakly suggest the hypothesis. • This improves efficiency of the system – preventing rules to be activated with little or no value. • A combining function can be used. Expert Systems: Principles and Programming, Fourth Edition

Difficulties with Certainty Factors • In MYCIN, which was very successful in diagnosis, there were difficulties with theoretical foundations of certain factors. • There was some basis for the CF values in probability theory and confirmation theory, but the CF values were partly ad hoc. • Also, the CF values could be the opposite of conditional probabilities. Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer Theory • The Dempster-Shafer Theory is a method of inexact reasoning. • It is based on the work of Dempster who attempted to model uncertainty by a range of probabilities rather than a single probabilistic number. Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • The Dempster-Shafer theory assumes that there is a fixed set of mutually exclusive and exhaustive elements called environment and symbolized by the Greek letter :  = {1, 2, …, N} Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • The environment is another term for the universe of discourse in set theory. • Consider the following: • = {rowboat, sailboat, destroyer, aircraft carrier} • These are all mutually exclusive elements Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • Consider the question: “What are the military boats?” • The answer would be a subset of  : {3, 4} = {destroyer, aircraft carrier} Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • Consider the question: “What boat is powered by oars?” • The answer would also be a subset of  : {1} = {rowboat} This set is called a singleton because it contains only one element. Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • Each of these subsets of  is a possible answer to the question, but there can be only one correct answer. • Consider each subset an implied proposition: • The correct answer is: {1, 2, 3) • The correct answer is: {1, 3} • All subsets of the environment can be drawn as a hierarchical lattice with  at the top and the null set  at the bottom Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • An environment is called a frame of discernment when its elements may be interpreted as possible answers and only one answer is correct. • If the answer is not in the frame, the frame must be enlarged to accommodate the additional knowledge of element.. Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • Mass Functions and Ignorance In Bayesian theory, the posterior probability changes as evidence is acquired. In Dempster-Shafer theory, the belief in evidence may vary. We talk about the degree of belief in evidence as analogous to the mass of a physical object – evidence measures the amount of mass. Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • Dempster-Shafer does not force belief to be assigned to ignorance – any belief not assigned to a subset is considered no belief (or non-belief) and just associated with the environment. • Every set in the power set of the environment which has mass > 0 is a focal element. • Every mass can be thought of as a function: m: P( )  [0, 1] Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • Combining Evidence Dempster’s rule combines mass to produce a new mass that represents the consensus of the original, possibly conflicting evidence The lower bound is called the support; the upper bound is called the plausibility; the belief measure is the total belief of a set and all its subsets. Expert Systems: Principles and Programming, Fourth Edition

Dempster-Shafer • The moving mass analogy is helpful to understanding the support and plausibility. • The support is the mass assigned to a set and all its subsets • Mass of a set can move freely into its subsets • Mass in a set cannot move into its supersets • Moving mass from a set into its subsets can only contribute to the plausibility of the subset, not its support. • Mass in the environment can move into any subset. Expert Systems: Principles and Programming, Fourth Edition

Approximate Reasoning • This is theory of uncertainty based on fuzzy logic and concerned with quantifying and reasoning using natural language where words have ambiguous meaning. • Fuzzy logic is a superset of conventional logic – extended to handle partial truth. • Soft-computing means computing not based on classical two-valued logics – includes fuzzy logic, neural networks, and probabilistic reasoning. Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Sets and Natural Language • A discrimination function is a way to represent which objects are members of a set. • 1 means an object is an element • 0 means an object is not an element • Sets using this type of representation are called “crisp sets” as opposed to “fuzzy sets”. • Fuzzy logic plays the middle ground – like human reasoning – everything consists of degrees – beauty, height, grace, etc. Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Sets and Natural Language • In fuzzy sets, an object may partially belong to a set measured by the membership function – grade of membership. • A fuzzy truth value is called a fuzzy qualifier. • Compatibility means how well one object conforms to some attribute. • There are many type of membership functions. • The crossover point is where  = 0.5 Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Set Operations • An ordinary crisp set is a special case of a fuzzy set with membership function [0, 1]. • All definitions, proofs, and theorems of fuzzy sets must be compatible in the limit as the fuzziness goes to 0 and the fuzzy sets become crisp sets. Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Set Operations Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Relations • A relation from a set A to a set B is a subset of the Cartesian product: A × B = {(a,b) | a  A and b  B} • If X and Y are universal sets, then R = {R(x, y) / (x, y) | (x, y)  X × Y} Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Relations • The composition of relations is the net effect of applying one relation after another. • For two binary relations P and Q, the composition of their relations is the binary relation: R(A, C) = Q(A, B)  P(B, C) Expert Systems: Principles and Programming, Fourth Edition

Table 5.7 Some Applications of Fuzzy Theory Expert Systems: Principles and Programming, Fourth Edition

Table 5.8 Some Fuzzy Terms of Natural Language Expert Systems: Principles and Programming, Fourth Edition

Linguistic Variables • One application of fuzzy sets is computational linguistics – calculating with natural language statements. • Fuzzy sets and linguistic variables can be used to quantify the meaning of natural language, which can then be manipulated. • Linguistic variables must have a valid syntax and semantics. Expert Systems: Principles and Programming, Fourth Edition

Extension Principle • The extension principle defines how to extend the domain of a given crisp function to include fuzzy sets. • Using this principle, ordinary or crisp functions can be extended to work a fuzzy domain with fuzzy sets. • This principle makes fuzzy sets applicable to all fields. Expert Systems: Principles and Programming, Fourth Edition

Fuzzy Logic • Just as classical logic forms the basis of expert systems, fuzzy logic forms the basis of fuzzy expert systems. • Fuzzy logic is an extension of multivalued logic – the logic of approximate reasoning – inference of possibly imprecise conclusions from a set of possibly imprecise premises. Expert Systems: Principles and Programming, Fourth Edition

Possibility and Probabilityand Fuzzy Logic • In fuzzy logic, possibility refers to allowed values. • Possibility distributions are not the same as probability distributions – frequency of expected occurrence of some random variable. Expert Systems: Principles and Programming, Fourth Edition

Translation Rules • Translation rules specify how modified or composite propositions are generated from their elementary propositions. 1. Type I modification rules 2. Type II composition rules 3. Type III quantification rules 4. Type IV quantification rules Expert Systems: Principles and Programming, Fourth Edition

State of UncertaintyCommercial Applications • There are two mountains – logic and uncertainty • Expert systems are built on the mountain of logic and must reach valid conclusions given a set of premises – valid conclusions given that – • The rules were written correctly • The facts upon which the inference engine generates valid conclusions are true facts • Today, fuzzy logic and Bayesian theory are most often used for uncertainty. Expert Systems: Principles and Programming, Fourth Edition

Summary • In this chapter, non-classical probability theories of uncertainty were discussed. • Certainty factors, Dempster-Shafer and fuzzy theory are ways of dealing with uncertainty in expert systems. • Certainty factors are simple to implement where inference chains are short (e.g. MYCIN) • Certainty factors are not generally valid for longer inference chains. Expert Systems: Principles and Programming, Fourth Edition

Chapter 5: Inexact Reasoning